Estimation of Production Functions: Fixed Effects in Panel Data

Lecture VIII

Analysis of Covariance

Looking at a representative regression model

It is well known that ordinary least squares (OLS) regressions of y on x and z are best linear unbiased estimators (BLUE) of α, β, and γ

* 1,

1,it it it ity x z u i N

t T

However, the results are corrupted if we do not observe z. Specifically if the covariance of x and z are correlated, then OLS estimates of the β are biased.

However, if repeated observations of a group of individuals are available (i.e., panel or longitudinal data) they may us to get rid of the effect of z.

For example if zit = zi (or the unobserved variable is the same for each individual across time), the effect of the unobserved variables can be removed by first-differencing the dependent and independent variables

, 1 , 1 , 1 , 1it i t it i t it i t it i ty y x x z z u u

, 1it i t iz z z

, 1 , 1 , 1 1,

2,

it i t it i t it i ty y x x u u i N

t T

Similarly if zit = zt (or the unobserved variables are the same for every individual at a any point in time) we can derive a consistent estimator by subtracting the mean of the dependent and independent variables for each individual

it i it i it i it iy y x x z z u u

it iz z

1

1

1

1

1

1

it i it i it i

T

i itt

T

i itt

T

i itt

y y x x u u

y yT

x xT

u uT

OLS estimators then provide unbiased and consistent estimates of β.

Unfortunately, if we have a cross-sectional dataset (i.e., T = 1) or a single time-series (i.e., N = 1) these transformations cannot be used.

Next, starting from the pooled estimates

Case I: Heterogeneous intercepts (αi ≠ α) and a homogeneous slope (βi = β).

*it i it ity x u

*it it ity x u

Case II: Heterogeneous slopes and intercepts (αi ≠ α , βi ≠ β )

*it i i it ity x u

Empirical Procedure

From the general model, we pose three different hypotheses: H1: Regression slope coefficients are

identical and the intercepts are not. H2: Regression intercepts are the same

and the slope coefficients are not. H3: Both slopes and the intercepts are the

same.

Estimation of different slopes and intercepts

1 1

1 1T T

i it i itt t

y y x xT T

, ,

, ,1 1

2

,1

ˆ ˆˆ 1,XX i XY i i i i i

T T

XX i it i it i XY i it i it it t

T

YY i it it

W W y x i N

W x x x x W x x y y

W y y

1, , , ,i YY i XY i XX i XY iRSS WW W W W

11

N

ii

S RSS

Covariance Matrices

X'X Nitrogen Phosphorous Potash X'Y beta alpha

Illinois

Nitrogen 1.2823 0.7194 1.5488 0.7415 0.7985 3.7917

Phosphorous 0.7160 0.6410 1.0156 0.2204-0.9813

Potash 1.5427 1.0174 2.0326 0.7894 0.2734

Indiana

Nitrogen 1.0346 0.2489 0.7220 0.6577 0.4386 3.6162

Phosphorous 0.2348 0.3717 0.2320 -0.0913-0.8905

Potash 0.7268 0.2448 0.6072 0.4587 0.5894

Pooled

Nitrogen 2.3168 0.9683 2.2708 1.3992 0.5924 3.9789

Phosphorous 0.9508 1.0128 1.2475 0.1291-0.9335 3.8851

Potash 2.2695 1.2622 2.6398 1.2481 0.4098

Estimation of different intercepts with the same slope

1 *

, ,1 1

,1

ˆ ˆ 1,W XX XY i i W i

N N

XX XX i XY XY ii i

N

YY YY ii

W W y x i N

W W W W

W W

12 YY XY XX XYS W W W W

Estimation of homogeneous slopes and intercepts

1 *

1 1

1 1

2

1 1

1 1 1 1

ˆ ˆˆ

1 1

XX XY

N T

XX it iti t

N T

XY it iti t

N T

YY iti t

N T N T

it iti t i t

T T y x

T x x x x

T x x y y

T y y

y y x xNT NT

13 YY XY XX XYS T T T T

Testing first for pooling both the slope and intercept terms:

* * *3 1 2

1 2

3 1

31

:

1 1~ 1 1 , 1

1

N

N

H

S SN K

F F N K NT N KSNT N K

If this hypothesis is rejected, we then test for homogeneity of the slopes, but heterogeneity of the constants

1 1 2

2 1

11

:

1~ 1 , 1

1

NH

S SN K

F F N K NT N KSNT N K

Dummy-Variable Formulation

1 1 1

2 2 2* * *1 2

3

0 0

0 0

0 0

N

N N

y x ue

y x ueY

y x ue

1 1 1 2 1 1

2 1 2 2 2 2

1 2

1

1 1 2

2

1 1 1

0 0

i i i Ki

i i i Kii i

it iT iT KiT

T

i T i i i iN

i i i T i j

y x x x

y x x xy x

y x x x

e M e

u M u u u u

E u E u u I E u u i j

Given this formulation, we know the OLS estimation of

The OLS estimation of α and β are obtained by minimizing

*it i it ity x u

* *

1 1

N N

i i i i i i i ii i

S u u y e x y e x

*

1 1

1 1 1 1

ˆ 1,

1 1

ˆ

i i i

T T

i it i iti t

N T N T

CV it i it i it i it ii t i t

y x i N

y y x xT T

x x x x x x y y

Sweeping the data

1TQ I eeT

1 1 1 11 4 4 4 41 0 0 0 1 1 1 11 1 1 110 1 0 0 1 1 1 1 4 4 4 41

40 0 1 0 1 1 1 1 1 1 1 114 4 4 40 0 0 1 1 1 1 1 1 1 1 114 4 4 4

1 2 3 4

1 2 3 4

4

1 2 3 4

1 2 3 4

1 1 2 3 4

2 1 2 3 4

3 1 2 3 4

4 1 2 3 4

1 1 1 11 4 4 4 4

1 1 1 114 4 4 414 1 1 1 114 4 4 4

1 1 1 114 4 4 4

14

14

14

14

x x x x

x x x xI ee x

x x x x

x x x x

x x x x x

x x x x x

x x x x x

x x x x x